Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/58

$AQ^'PQA$. But the line-integral of the closed path is zero, therefore those of the two paths are equal.

Hence if the potential is given at any one point of such a region, that at any other point is determinate.

20.] Theorem II. In a cyclic region in which the equation

$Xdx+Ydy + Zdz = -D\Psi$

is everywhere fulfilled, the line-integral from, $A$ to $P$, along a line drawn within the region, will not in general be determinate unless the channel of communication between $A$ and $P$ be specified

Let $K$ be the cyclomatic number of the region, then $K$ sections of the region may be made by surfaces which we may call Diaphragms, so as to close up $K$ of the channels of communication, and reduce the region to an acyclic condition without destroying its continuity.

The line-integral from $A$ to any point $P$ taken along a line which does not cut any of these diaphragms will be, by the last theorem, determinate in value.

Now let $A$ and $P$ be taken indefinitely near to each other, but on opposite sides of a diaphragm, and let $K$ be the line-integral from $A$ to $P$.

Let $A^'$ and $P^'$ be two other points on opposite sides of the same diaphragm and indefinitely near to each other, and let $K^'$ be the line-integral from $A^'$ to $P^'$. Then $K^' = K$.

For if we draw $A A^'$ and $P P^'$, nearly coincident, but on opposite sides of the diaphragm, the line-integrals aloug these lines will be equal. Suppose each equal to $Z$, then the line-integral of $A^' P^'$ is equal to that of $A^' A + A P+P P^'= -L + K+L = K =$ that of $A P$.

Hence the line-integral round a closed curve which passes through one diaphragm of the system in a given direction is a constant quantity $K$. This quantity is called the Cyclic constant corresponding to the given cycle.

Let any closed curve be drawn within the region, and let it cut the diaphragm of the first cycle $p$ times in the positive direction and p times in the negative direction, and let $p-p^' = n_1$ . Then the line-integral of the closed curve will be $n_1 K_1$.

Similarly the line-integral of any closed curve will be

$n_1 K_1 +n_2 K_2 + ... +n_k K_k$;

where $n_K$ represents the excess of the number of positive passages of the curve through the diaphragm of the cycle $K$ over the number of negative passages.